The competitive hybridization of two probes can be described by equations

1 and

2, where

*P*,

*T*, and

*H* are the probe, target, and probe/target hybrid, respectively (subscripts 1 and 2 indicate the competing probes). In FISH, the probe and target can have folded and unfolded conformations (

47), both of which are included in the definitions of

*P* and

*T* in equations

1 and

2 but are not shown for the sake of simplicity. If probes are added in excess of the target {i.e., [

*P*_{1}]

_{0} ≈ [

*P*_{2}]

_{0} [

*T*]

_{0}}, then [

*P*_{1}]

_{0} = [

*P*_{1}] + [

*H*_{1}] ≈ [

*P*_{1}] and [

*P*_{2}]

_{0} = [

*P*_{2}] + [

*H*_{2}] ≈ [

*P*_{2}], where [

*P*_{1}]

_{0} and [

*P*_{2}]

_{0} are the initial probe concentrations in the hybridization buffer. Thus, the equilibrium constants,

*K*_{1} and

*K*_{2}, of the hybridizations can be described by equations

3 and

4. These constants correspond to the overall equilibrium constants of the main processes that occur during in situ hybridization (

47). Dividing equation

3 by equation

4 and rearranging result in equation

5, which describes the ratio of the hybrids.

In competitive hybridizations, the experimental response is obtained as the fluorescence of one of the probes. Let this probe be probe

*P*_{1}. To link the response of

*P*_{1} to ribosome quantity, we started with the mass balance of probe and target as described by equations

6 and

7, where [

*P*_{T}] and [

*T*_{T}] are the total probe and target concentrations, respectively. The partitioning of total target into free (

*T*) and hybridized (

*H*_{1} and

*H*_{2}) forms (equation

7) does not always allow a unique solution for the intended derivation of ribosome quantity from the experimental response. However, in the ideal case, when the rRNA targets are nearly saturated {i.e., [

*T*] ≈ 0; hence, [

*T*]

[

*H*_{1}] + [

*H*_{2}]}, equation

7 simplifies to equation

8. Then it is possible to determine the fraction of targets hybridized to the fluorescent probe ([

*H*_{1}]/[

*T*_{T}]) as a function of equilibrium constants and probe concentrations only, as shown in equation

9. The rightmost term of this equation is obtained by substituting [

*H*_{1}] and [

*H*_{2}] from the corresponding expressions in equations

3 and

4.

Equation

9 can be rearranged in two steps. First, dividing both the numerator and the denominator on the rightmost side by [

*P*_{T}] results in equation

10, where

*R* is the probe ratio (i.e.,

*R*_{i} = [

*P*_{i}]

_{0}/[

*P*_{T}]). Second, equation

11 is obtained by substituting

*R*_{2} = 1 −

*R*_{1} in equation

10, dividing the nominator and denominator by

*K*_{2}, and rearranging. This equation allows quantification of rRNA targets from competitive hybridization experiments if the

*K*_{1}/

*K*_{2} ratio is known.

Finally, assuming that fluorescence intensity (

*F*) is proportional to the fraction of targets hybridized to the fluorescent probe (i.e., [

*H*_{1}]/[

*T*_{T}]), it can be described by equation

12, where β is a proportionality factor and

is the background fluorescence. Combining equations

11 and

12 results in equation

13, which allows quantification of the

*K*_{1}/

*K*_{2} ratio from competitive hybridization experiments. However, the critical assumption that nearly all rRNAs are hybridized with probes (equation

8) must be fulfilled when equations

11 and

13 are used. This assumption is true if two conditions are met: (i) the amount of at least one probe is greater than the amount of the target, and (ii) this probe is at the high-fluorescence plateau of its dissociation profile.